| L(s) = 1 | + 2.82i·2-s − 8.00·4-s − 21.5i·5-s − 22.6i·8-s + 60.9·10-s + 57.9i·11-s + 304.·13-s + 64.0·16-s + 323. i·17-s − 152.·19-s + 172. i·20-s − 164·22-s − 538. i·23-s + 161.·25-s + 861. i·26-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s − 0.861i·5-s − 0.353i·8-s + 0.609·10-s + 0.479i·11-s + 1.80·13-s + 0.250·16-s + 1.11i·17-s − 0.421·19-s + 0.430i·20-s − 0.338·22-s − 1.01i·23-s + 0.257·25-s + 1.27i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.114999739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.114999739\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2.82iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 21.5iT - 625T^{2} \) |
| 11 | \( 1 - 57.9iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 304.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 323. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 152.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 538. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 83.4iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 761.T + 9.23e5T^{2} \) |
| 37 | \( 1 + 600T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.61e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.16e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 1.07e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.10e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 3.66e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 2.58e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 8.27e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 1.83e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 9.29e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 3.08e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 1.35e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 2.04e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 761.T + 8.85e7T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.366687416013063118963813301747, −8.492162621931110661561905947897, −8.316024713443951555472696221844, −6.97155461498410949225728773605, −6.20148479236185417276057188814, −5.37406923649013552214323095183, −4.36416300899380324020028220677, −3.60742442495824553072968936075, −1.82295189304410097979574740647, −0.74760257563252851908536910072,
0.71708373298308893948165357423, 1.90496888449614744469087518679, 3.18284466812979469483569494879, 3.64799280207427927223877316871, 4.99671166967939936994122519644, 6.03515767951542469027152161018, 6.84186939513581472710495659672, 7.928136092396495692188792679467, 8.790525644409382533098047131679, 9.559922384369155888937866016129
